The interactive film is in the book's Companion

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The Polyak-Łojasiewicz Condition

Chapter 12 · Mathematics for AI and Machine Learning · Advanced · 1:55

An interactive live film: a non-convex loss that satisfies the PL inequality, both sides of the inequality live at a point you drag, and gradient descent's gap falling geometrically on a log-scale chart.

Narration

Here is a loss that is not convex: f of w equals w zero squared, plus three times sine squared of w zero, plus w one squared.

Over the purple bands on the floor, the surface curves downward along w zero. Still, its only critical point is the minimum at the origin, where f star is zero.

The rest of the narration is in the book's Companion.

More from Chapter 12

Interactive1:08Gradient Descent in an Ill-Conditioned ValleyChapter 12Preview1:28The Descent Lemma: A Bowl That Bounds Every StepChapter 12Preview1:33Heavy-Ball Momentum: Zig-Zags Cancel, the Valley Floor Adds UpChapter 12Preview1:43A Large Step Selects a Flatter MinimumChapter 12Preview2:12Sophia: Newton Steps with a ClipChapter 12Preview2:14Adam and AdamW: A Step Size for Every CoordinateChapter 12Preview1:55Backtracking Line Search and the Armijo ConditionChapter 12Preview1:50Proximal Gradient and Soft-Thresholding: Exact ZerosChapter 12Preview1:53KKT Conditions: the Saddle Point of the LagrangianChapter 12Preview1:52Mirror Descent Stays Inside the SimplexChapter 12